This is a rhombicosidodecahedron, one of the Archimedean solids.
If one pentagonal cupola is removed from this polyhedron, the result is the diminished rhombicosidodecahedron, which is one of the Johnson solids (J76).
The next step is to take another J76, and attach it to the first one, so that their decagonal faces meet.
I’m calling the result the “double rhombicosidodecahedron.”
I did these manipulations of polyhedra and their images with a program called Stella 4d: Polyhedron Navigator. There’s a free trial download available, if you’d like to try the program for yourself, and it’s at this website.
I made this polyhedron by creating a zonohedron based on the edges and faces of the truncated tetrahedron. Only the blue hexagons are irregular. Stella 4d was used in its creation, and you may try this program for free at http://www.software3d.com/Stella.php.
In yesterday’s post, I unveiled my annual birthday star for my new age, 54. Today, I’m placing that 54-pointed star on each of the thirty faces of a rhombic triacontahedron. I use a program called Stella 4d (free trial available right here) to do this, and it allows images on polyhedron-faces to either be placed inside the face, or around the face. Here’s the “inside” version: